Geodesic Laminations Research Articles

The Liouville current of holomorphic quadratic differential metrics

In this paper we study the Liouville current of flat cone metrics coming from a holomorphic quadratic differential. Anja Bankovic and Christopher J. Leininger proved in Bankovic and Leininger (Trans Am Math Soc 370:1867–1884, 2018) that for a fixed closed surface, there is an injection map from the space of flat cone metrics to the space of geodesic currents. We manage to show that metrics coming from holomorphic quadratic differentials can be distinguished from other flat metrics by just looking at the geodesic currents. The key idea is to analyze the support of Liouville current, which is a topological invariant independent of the metric, and get information about cone angles and holonomy. The holonomy part involves some subtlety of relationship between singular foliation and geodesic lamination. We also obtain a new proof of a classical result that almost all simple geodesics of a quadratic differential metric will be dense in the surface. Furthermore, for other flat cone metrics, there is no simple dense geodesic.

Earthquakes on the Once-Punctured Torus

We study earthquake deformations on Teichmüller space associated with simple closed curves of the once-punctured torus. We describe two methods to get an explicit form of the earthquake deformation for any simple closed curve. The first method is rooted in linear recurrence relations, the second in hyperbolic geometry. The two methods align, providing both an algebraic and geometric interpretation of the earthquake deformations. We convert the expressions to other coordinate systems for Teichmüller space to examine earthquake deformations further. Two families of curves are used as examples. Examining the limiting behavior of each gives insight into earthquakes about measured geodesic laminations, of which simple closed curves are a special case.

Laminations and 2-filling rays on infinite type surfaces

The loop graph of an infinite type surface is an infinite diameter hyperbolic graph first studied in detail by Juliette Bavard. An important open problem in the study of infinite type surfaces is to describe the boundary of the loop graph as a space of geodesic laminations. We approach this problem by constructing the first examples of 2-filling rays on infinite type surfaces. Such rays have strong filling properties while failing to correspond to points on the boundary of the loop graph. As such they may be thought of as “fake boundary points.” We give multiple constructions using both a hands-on combinatorial approach and an approach using train tracks and automorphisms of flat surfaces. In addition, our approaches are sufficiently robust to describe all 2-filling rays with certain other basic properties as well as to produce uncountably many distinct mapping class group orbits.

Counting hyperbolic multigeodesics with respect to the lengths of individual components and asymptotics of Weil–Petersson volumes

Given a connected, oriented, complete, finite area hyperbolic surface $X$ of genus $g$ with $n$ punctures, Mirzakhani showed that the number of multi-geodesics on $X$ of total hyperbolic length $\leq L$ in the mapping class group orbit of a given simple or filling closed multi-curve is asymptotic as $L \to \infty$ to a polynomial in $L$ of degree $6g-6+2n$. We establish asymptotics of the same kind for countings of multi-geodesics in mapping class group orbits of simple or filling closed multi-curves that keep track of the hyperbolic lengths of individual components, proving and generalizing a conjecture of Wolpert. In the simple case we consider more precise countings that also keep track of the class of the multi-geodesics in the space of projective measured geodesic laminations. We provide a unified geometric and topological description of the leading terms of the asymptotics of all the countings considered. Our proofs combine techniques and results from several papers of Mirzakhani as well as ideas introduced by Margulis in his thesis.

A Closed Subgroup of the Homeomorphism Group of the Circle with Property (T)

Abstract We exhibit a topological group $G$ with property (T) acting nonelementarily and continuously on the circle. This group is an uncountable totally disconnected closed subgroup of ${\operatorname {Homeo}}^+(\textbf {S}^1)$. It has a large unitary dual since it separates points. It comes from homeomorphisms of dendrites and a kaleidoscopic construction. Alternatively, it can be seen as the group of elements preserving some specific geodesic lamination of the hyperbolic disk. We also prove that this action is unique up to conjugation and that it cannot be smoothened in any way. Finally, we determine the universal minimal flow of the group $G$.

Train tracks and measured laminations on infinite surfaces

Let X X be an infinite Riemann surface equipped with its conformal hyperbolic metric such that the action of the fundamental group π 1 ( X ) \pi _1(X) on the universal covering X ~ \tilde {X} is of the first kind. We first prove that any geodesic lamination on X X is nowhere dense. Given a fixed geodesic pants decomposition of X X we define a family of train tracks on X X such that any geodesic lamination on X X is weakly carried by at least one train track. The set of measured laminations on X X carried by a train track is in a one to one correspondence with the set of edge weight systems on the train track. Furthermore, the above correspondence is a homeomorphism when we equipped the measured laminations (weakly carried by a train track) with the weak* topology and the edge weight systems with the topology of pointwise (weak) convergence. The space M L b ( X ) ML_b(X) of bounded measured laminations appears prominently when studying the Teichmüller space T ( X ) T(X) of X X . If X X has a bounded pants decomposition, a measured lamination on X X weakly carried by a train track is bounded if and only if the corresponding edge weight system has a finite supremum norm. The space M L b ( X ) ML_b(X) is equipped with the uniform weak* topology. The correspondence between bounded measured laminations weakly carried by a train track and their edge weight systems is a homeomorphism for the uniform weak* topology on M L b ( X ) ML_b(X) and the topology induced by supremum norm on the edge weight system.

Bending laminations on convex hulls of anti‐de Sitter quasi‐circles

Let $\lambda_-$ and $\lambda_+$ be two bounded measured laminations on the hyperbolic disk $\mathbb H^2$, which "strongly fill" (definition below). We consider the left earthquakes along $\lambda_-$ and $\lambda_+$, considered as maps from the universal Teichm\"uller space $\mathcal T$ to itself, and we prove that the composition of those left earthquakes has a fixed point. The proof uses anti-de Sitter geometry. Given a quasi-symmetric homeomorphism $u:{\mathbb RP}^1\to {\mathbb RP}^1$, the boundary of the convex hull in $AdS^3$ of its graph in ${\mathbb RP}^1\times{\mathbb RP}^1\simeq \partial AdS^3$ is the disjoint union of two embedded copies of the hyperbolic plane, pleated along measured geodesic laminations. Our main result is that any pair of bounded measured laminations that "strongly fill" can be obtained in this manner.

A Thurston boundary for infinite-dimensional Teichmüller spaces

For a compact surface $X_0$, Thurston introduced a compactification of its Teichm\"uller space $\mathcal T(X_0)$ by completing it with a boundary $\mathcal{PML}(X_0)$ consisting of projective measured geodesic laminations. We introduce a similar bordification for the Teichm\"uller space $\mathcal T(X_0)$ of a noncompact Riemann surface $X_0$, using the technical tool of geodesic currents. The lack of compactness requires the introduction of certain uniformity conditions which were unnecessary for compact surfaces. A technical step, providing a convergence result for earthquake paths in $\mathcal T(X_0)$, may be of independent interest.

Laminational Models for Some Spaces of Polynomials of Any Degree

The so-called pinched model of the Mandelbrot set is due to A.~Douady, J.~H.~Hubbard and W.~P.~Thurston. It can be described in the language of geodesic laminations. The combinatorial model is the quotient space of the unit disk under an equivalence relation that, loosely speaking, pinches the disk in the plane (whence the name of the model). The significance of the model lies in particular in the fact that this quotient is planar and therefore can be easily visualized. The conjecture that the Mandelbrot set is actually homeomorphic to this model is equivalent to the celebrated MLC conjecture stating that the Mandelbrot set is locally connected. For parameter spaces of higher degree polynomials no combinatorial model is known. One possible reason may be that the higher degree analog of the MLC conjecture is known to be false. We investigate to which extent a geodesic lamination is determined by the location of its critical sets and when different choices of critical sets lead to essentially the same lamination. This yields models of various parameter spaces of laminations similar to the pinched model of the Mandelbrot set.

SQUARE-INTEGRABILITY OF THE MIRZAKHANI FUNCTION AND STATISTICS OF SIMPLE CLOSED GEODESICS ON HYPERBOLIC SURFACES

Given integers$g,n\geqslant 0$satisfying$2-2g-n<0$, let${\mathcal{M}}_{g,n}$be the moduli space of connected, oriented, complete, finite area hyperbolic surfaces of genus$g$with$n$cusps. We study the global behavior of the Mirzakhani function$B:{\mathcal{M}}_{g,n}\rightarrow \mathbf{R}_{{\geqslant}0}$which assigns to$X\in {\mathcal{M}}_{g,n}$the Thurston measure of the set of measured geodesic laminations on$X$of hyperbolic length${\leqslant}1$. We improve bounds of Mirzakhani describing the behavior of this function near the cusp of${\mathcal{M}}_{g,n}$and deduce that$B$is square-integrable with respect to the Weil–Petersson volume form. We relate this knowledge of$B$to statistics of counting problems for simple closed hyperbolic geodesics.

Endperiodic automorphisms of surfaces and foliations

We extend the unpublished work of Handel and Miller on the classification, up to isotopy, of endperiodic automorphisms of surfaces. We give the Handel–Miller construction of the geodesic laminations, give an axiomatic theory for pseudo-geodesic laminations, show that the geodesic laminations satisfy the axioms, and prove that pseudo-geodesic laminations satisfying our axioms are ambiently isotopic to the geodesic laminations. The axiomatic approach allows us to show that the given endperiodic automorphism is isotopic to a smooth endperiodic automorphism preserving smooth laminations ambiently isotopic to the original ones. Using the axioms, we also prove the ‘transfer theorem’ for foliations of 3-manifolds, namely that, if two depth-one foliations${\mathcal{F}}$and${\mathcal{F}}^{\prime }$are transverse to a common one-dimensional foliation${\mathcal{L}}$whose monodromy on the non-compact leaves of${\mathcal{F}}$exhibits the nice dynamics of Handel–Miller theory, then${\mathcal{L}}$also induces monodromy on the non-compact leaves of${\mathcal{F}}^{\prime }$exhibiting the same nice dynamics. Our theory also applies to surfaces with infinitely many ends.

Limits of harmonic maps and crowned hyperbolic surfaces

We consider harmonic diffeomorphisms to a fixed hyperbolic target Y Y from a family of domain Riemann surfaces degenerating along a Teichmüller ray. We use the work of Minsky to show that there is a limiting harmonic map from the conformal limit of the Teichmüller ray to a crowned hyperbolic surface. The target surface is the metric completion of the complement of a geodesic lamination on Y Y . The conformal limit is obtained by attaching half-planes and cylinders to the critical graph of the holomorphic quadratic differential determining the ray. As an application, we provide a new proof of the existence of harmonic maps from any punctured Riemann surface to a given crowned hyperbolic target of the same topological type.

Bijections of geodesic lamination space preserving left Hausdorff convergence

We introduce an asymmetric distance function, which we call the “left Hausdorff distance function”, on the space of geodesic laminations on a closed hyperbolic surface of genus at least 2. This distance is an asymmetric version of the Hausdorff distance between compact subsets of a metric space. We prove a rigidity result for the action of the extended mapping class group of the surface on the space of geodesic laminations equipped with the topology induced from this distance. More specifically, we prove that there is a natural homomorphism from the extended mapping class group into the group of bijections of the space of geodesic laminations that preserve left Hausdorff convergence and that this homomorphism is an isomorphism.

Representation of spaciously situated perfect attractors of diffeomorphisms by geodesic laminations

Representation of spaciously situated perfect attractors of diffeomorphisms by geodesic laminations

Topological laminations on surfaces

We present and prove a topological characterization of geodesic laminations on hyperbolic surfaces of finite type.

Hamiltonian properties of earthquake flows on surfaces with closed geodesic boundary

The Teichmuller space $${\mathcal {T}}_S(\mathbf {b})$$ of hyperbolic metrics on a surface S with fixed lengths at the boundary components is symplectic. We prove that any sum of infinitesimal earthquakes on S that is tangent to $${\mathcal {T}}_S(\mathbf {b})$$ is Hamiltonian, by providing a Hamiltonian $$\mathbb {L}$$ . Such function extends the classical length map associated to a compactly supported measured geodesic lamination and shares with it some peculiar properties, such as properness and strict convexity along earthquakes paths under usual topological conditions. As an application, we prove that any non-Fuchsian affine representation of $$\pi _1(S)$$ into $$\mathbb {R}^{2,1}\rtimes SO_0(2,1)$$ with cocompact discrete linear part is determined by the singularities of the two invariant regular domains in $$\mathbb {R}^{2,1}$$ pointed out by Barbot, once the boundary lengths are fixed.

On a problem of A. Weil

A topological invariant of the geodesic laminations on a modular surface is constructed. The invariant has a continuous part (the tail of a continued fraction) and a combinatorial part (the singularity data). It is shown, that the invariant is complete, i.e. the geodesic lamination can be recovered from the invariant. The continuous part of the invariant has geometric meaning of a slope of lamination on the surface.

On the volume of Anti-de Sitter maximal globally hyperbolic three-manifolds

We study the volume of maximal globally hyperbolic Anti-de Sitter manifolds containing a closed orientable Cauchy surface S, in relation to some geometric invariants depending only on the two points in Teichmüller space of S provided by Mess’ parameterization—namely on two isotopy classes of hyperbolic metrics h and h′ on S. The main result of the paper is that the volume coarsely behaves like the minima of the L 1-energy of maps from (S, h) to (S, h′). The study of L p -type energies had been suggested by Thurston, in contrast with the well-studied Lipschitz distance. A corollary of our result shows that the volume of maximal globally hyperbolic Anti-de Sitter manifolds is bounded from above by the exponential of (any of the two) Thurston’s Lipschitz asymmetric distances, up to some explicit constants. Although there is no such bound from below, we provide examples in which this behavior is actually realized. We prove instead that the volume is bounded from below by the exponential of the Weil–Petersson distance. The proof of the main result uses more precise estimates on the behavior of the volume, which is proved to be coarsely equivalent to the length of the (left or right) measured geodesic lamination of earthquake from (S, h) to (S, h′), and to the minima of the holomorphic 1-energy.

Maximally stretched laminations on geometrically finite hyperbolic manifolds

Let Gamma_0 be a discrete group. For a pair (j,rho) of representations of Gamma_0 into PO(n,1)=Isom(H^n) with j geometrically finite, we study the set of (j,rho)-equivariant Lipschitz maps from the real hyperbolic space H^n to itself that have minimal Lipschitz constant. Our main result is the existence of a geodesic lamination that is "maximally stretched" by all such maps when the minimal constant is at least 1. As an application, we generalize two-dimensional results and constructions of Thurston and extend his asymmetric metric on Teichm\"uller space to a geometrically finite setting and to higher dimension. Another application is to actions of discrete subgroups Gamma of PO(n,1)xPO(n,1) on PO(n,1) by right and left multiplication: we give a double properness criterion for such actions, and prove that for a large class of groups Gamma the action remains properly discontinuous after any small deformation of Gamma inside PO(n,1)xPO(n,1).

Hitchin characters and geodesic laminations

For a closed surface S, the Hitchin component Hit_n(S) is a preferred component of the character variety consisting of group homomorphisms from the fundamental group pi_1(S) to the Lie group PSL_n(R). We construct a parametrization of the Hitchin component that is well-adapted to a maximal geodesic lamination on the surface. This is a natural extension of Thurston's parametrization of the Teichmueller space of S by shear coordinates associated to a maximal geodesic lamination, corresponding to the case n=2. However, significantly new ideas are needed in this higher dimensional case. The article concludes with a few applications.